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204A-slides02b

# 204A-slides02b - (2b-P.1 Applications of Growth Theory I...

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(2b)-P.1 Applications of Growth Theory I: Growth Accounting Objective: Use empirical data on output, capital stocks, and labor supply, to interpret history (“accounting”), to compare across countries, or to make projections. • Data sets: observations (Y t , K t , L t ) at discrete dates t. - Productivity index A is not directly observable – must be inferred. - Though capital and labor shares vary, commonly assume Cobb-Douglas: α k ( k ) α . - Common labels: Y/L = “labor productivity” vs. A = “total/multi factor productivity” • Growth accounting with Cobb-Douglas production: - Write Y = K α ( AL ) 1 α = R K α L 1 α with R = A 1 α = the Solow Residual => ln Y = α ln K + (1 α )ln L + ln R with ln R = (1 α )ln A - Take time-differences to approximate growth [ math: d ln( x )/ dt = 1 x dx dt ≈ Δ ln( x ) ] => Δ ln Y = α ⋅Δ ln K + (1 α ) ⋅Δ ln L + Δ ln R - General notation for growth over an interval: g x = 1 t 1 t 0 [ln( x ( t 1 ) ln( x ( t 0 )] => g Y = α g K + (1 α ) g L + g R

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(2b)-P.2 Decomposing Growth Rates • Key equations: g Y = α g K + (1 α ) g L + g R = α g K + (1 α ) g L + (1 α ) g A • From the data: compute ( g Y , g K , g L ) for various time periods; estimate α . - Use Cobb-Douglas to infer g R = g Y α g K (1 α ) g L or R = Y ( K α L 1 α ) - Use R = A 1 α to infer TFP: g A = g R /(1 α ) or A = R 1/(1 α ) • Growth accounting for per-capita variables [Math: Growth of ratio = difference of rates] - Income: g Y / L = g Y g L = α g K α g L + g R = α g K / L + g R - Growth in K/L called capital deepening . Conclude: Growth of per-capita income = Capital deepening and productivity growth. - Growth accounting means decomposition of growth rates into components. - Sometimes decompose further: multiple types of capital and labor, each weighted by its factor share. Multi-factor productivity is the residual – a “measure of ignorance.”
(2b)-P.3 Approximation for General Production Functions • Claim: Growth accounting formulas are generally valid as approximations: - Argument: Y = f ( k ) AL = f ( e x ) AL with x = ln( k ) = ln( K AL ) => ln Y = ln f ( e x ) + ln A + ln L Note that d dt ln f ( e x ) [ ] = f '( e x ) f ( e x ) e x = f '( k ) f ( k ) k = α k ( k ) => d dt ln Y = α k ( k ) d ln k dt + d dt ln A + d dt ln L = α k ( k ) d dt ln K + (1 α k ( k ))( d dt ln A + d dt ln L ) - Conclude:

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204A-slides02b - (2b-P.1 Applications of Growth Theory I...

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